On the number of invariant straight lines for polynomial differential systems
On the number of invariant straight lines for polynomial differential systems
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DOI：
10.2140/pjm.1998.184.207
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发表时间：
1998-06
影响因子：
0.6
通讯作者：
Joan C. Artés;B. Grünbaum;J. Llibre
中科院分区：
文献类型：
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作者：
Joan C. Artés;B. Grünbaum;J. Llibre
如果P和Q是实变量x和y中的两个实多项式，且P2+Q的次数为2n，则称多项式微分系统x‘=P(x，y)，y’=Q(x，y)具有n次.设α(N)是n次n>；1次多项式微分系统具有有限多条不变直线的最大不变直线个数.上世纪80年代，S的猜想在从事多项式微分系统的数学家中流传开来。猜想：如果n是偶数，α(N)是2n+1，如果n是奇数，α(N)是2n+2。建立了n=2，3，4的猜想.在本文中，我们证明了猜想对n>；4不成立.具体地说，我们证明了α(5)=14.此外，我们还给出了n∈{6，7，.。。，20}。我们还证明了如果n是偶数，则2n+1≤α(N)≤3n−1；如果n是奇数，则2n+2≤α(N)≤3n−1。
If P and Q are two real polynomials in the real variables x and y such that the degree of P 2 +Q is 2n, then we say that the polynomial differential system x′ = P (x, y), y′ = Q(x, y) has degree n. Let α(n) be the maximum number of invariant straight lines possible in a polynomial differential systems of degree n > 1 having finitely many invariant straight lines. In the 1980’s the following conjecture circulated among mathematicians working in polynomial differential systems. Conjecture: α(n) is 2n+ 1 if n is even, and α(n) is 2n+ 2 if n is odd. The conjecture was established for n = 2, 3, 4. In this paper we prove that, in general, the conjecture is not true for n > 4. Specifically, we prove that α(5) = 14. Moreover, we present counterexamples to the conjecture for n ∈ {6, 7, . . . , 20}. We also show that 2n + 1 ≤ α(n) ≤ 3n − 1 if n is even, and that 2n+ 2 ≤ α(n) ≤ 3n− 1 if n is odd.